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# Polytope of Type {45,6}

Atlas Canonical Name : {45,6}*540
if this polytope has a name.
Group : SmallGroup(540,52)
Rank : 3
Schlafli Type : {45,6}
Number of vertices, edges, etc : 45, 135, 6
Order of s0s1s2 : 90
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{45,6,2} of size 1080
{45,6,3} of size 1620
Vertex Figure Of :
{2,45,6} of size 1080
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {45,2}*180, {15,6}*180
5-fold quotients : {9,6}*108
9-fold quotients : {15,2}*60
15-fold quotients : {9,2}*36, {3,6}*36
27-fold quotients : {5,2}*20
45-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {90,6}*1080b
3-fold covers : {45,18}*1620, {45,6}*1620a, {135,6}*1620
Permutation Representation (GAP) :
```s0 := (  2,  3)(  4, 13)(  5, 15)(  6, 14)(  7, 10)(  8, 12)(  9, 11)( 16, 32)
( 17, 31)( 18, 33)( 19, 44)( 20, 43)( 21, 45)( 22, 41)( 23, 40)( 24, 42)
( 25, 38)( 26, 37)( 27, 39)( 28, 35)( 29, 34)( 30, 36)( 46, 91)( 47, 93)
( 48, 92)( 49,103)( 50,105)( 51,104)( 52,100)( 53,102)( 54,101)( 55, 97)
( 56, 99)( 57, 98)( 58, 94)( 59, 96)( 60, 95)( 61,122)( 62,121)( 63,123)
( 64,134)( 65,133)( 66,135)( 67,131)( 68,130)( 69,132)( 70,128)( 71,127)
( 72,129)( 73,125)( 74,124)( 75,126)( 76,107)( 77,106)( 78,108)( 79,119)
( 80,118)( 81,120)( 82,116)( 83,115)( 84,117)( 85,113)( 86,112)( 87,114)
( 88,110)( 89,109)( 90,111);;
s1 := (  1, 64)(  2, 66)(  3, 65)(  4, 61)(  5, 63)(  6, 62)(  7, 73)(  8, 75)
(  9, 74)( 10, 70)( 11, 72)( 12, 71)( 13, 67)( 14, 69)( 15, 68)( 16, 49)
( 17, 51)( 18, 50)( 19, 46)( 20, 48)( 21, 47)( 22, 58)( 23, 60)( 24, 59)
( 25, 55)( 26, 57)( 27, 56)( 28, 52)( 29, 54)( 30, 53)( 31, 80)( 32, 79)
( 33, 81)( 34, 77)( 35, 76)( 36, 78)( 37, 89)( 38, 88)( 39, 90)( 40, 86)
( 41, 85)( 42, 87)( 43, 83)( 44, 82)( 45, 84)( 91,109)( 92,111)( 93,110)
( 94,106)( 95,108)( 96,107)( 97,118)( 98,120)( 99,119)(100,115)(101,117)
(102,116)(103,112)(104,114)(105,113)(121,125)(122,124)(123,126)(127,134)
(128,133)(129,135)(130,131);;
s2 := ( 46, 91)( 47, 92)( 48, 93)( 49, 94)( 50, 95)( 51, 96)( 52, 97)( 53, 98)
( 54, 99)( 55,100)( 56,101)( 57,102)( 58,103)( 59,104)( 60,105)( 61,106)
( 62,107)( 63,108)( 64,109)( 65,110)( 66,111)( 67,112)( 68,113)( 69,114)
( 70,115)( 71,116)( 72,117)( 73,118)( 74,119)( 75,120)( 76,121)( 77,122)
( 78,123)( 79,124)( 80,125)( 81,126)( 82,127)( 83,128)( 84,129)( 85,130)
( 86,131)( 87,132)( 88,133)( 89,134)( 90,135);;
poly := Group([s0,s1,s2]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(135)!(  2,  3)(  4, 13)(  5, 15)(  6, 14)(  7, 10)(  8, 12)(  9, 11)
( 16, 32)( 17, 31)( 18, 33)( 19, 44)( 20, 43)( 21, 45)( 22, 41)( 23, 40)
( 24, 42)( 25, 38)( 26, 37)( 27, 39)( 28, 35)( 29, 34)( 30, 36)( 46, 91)
( 47, 93)( 48, 92)( 49,103)( 50,105)( 51,104)( 52,100)( 53,102)( 54,101)
( 55, 97)( 56, 99)( 57, 98)( 58, 94)( 59, 96)( 60, 95)( 61,122)( 62,121)
( 63,123)( 64,134)( 65,133)( 66,135)( 67,131)( 68,130)( 69,132)( 70,128)
( 71,127)( 72,129)( 73,125)( 74,124)( 75,126)( 76,107)( 77,106)( 78,108)
( 79,119)( 80,118)( 81,120)( 82,116)( 83,115)( 84,117)( 85,113)( 86,112)
( 87,114)( 88,110)( 89,109)( 90,111);
s1 := Sym(135)!(  1, 64)(  2, 66)(  3, 65)(  4, 61)(  5, 63)(  6, 62)(  7, 73)
(  8, 75)(  9, 74)( 10, 70)( 11, 72)( 12, 71)( 13, 67)( 14, 69)( 15, 68)
( 16, 49)( 17, 51)( 18, 50)( 19, 46)( 20, 48)( 21, 47)( 22, 58)( 23, 60)
( 24, 59)( 25, 55)( 26, 57)( 27, 56)( 28, 52)( 29, 54)( 30, 53)( 31, 80)
( 32, 79)( 33, 81)( 34, 77)( 35, 76)( 36, 78)( 37, 89)( 38, 88)( 39, 90)
( 40, 86)( 41, 85)( 42, 87)( 43, 83)( 44, 82)( 45, 84)( 91,109)( 92,111)
( 93,110)( 94,106)( 95,108)( 96,107)( 97,118)( 98,120)( 99,119)(100,115)
(101,117)(102,116)(103,112)(104,114)(105,113)(121,125)(122,124)(123,126)
(127,134)(128,133)(129,135)(130,131);
s2 := Sym(135)!( 46, 91)( 47, 92)( 48, 93)( 49, 94)( 50, 95)( 51, 96)( 52, 97)
( 53, 98)( 54, 99)( 55,100)( 56,101)( 57,102)( 58,103)( 59,104)( 60,105)
( 61,106)( 62,107)( 63,108)( 64,109)( 65,110)( 66,111)( 67,112)( 68,113)
( 69,114)( 70,115)( 71,116)( 72,117)( 73,118)( 74,119)( 75,120)( 76,121)
( 77,122)( 78,123)( 79,124)( 80,125)( 81,126)( 82,127)( 83,128)( 84,129)
( 85,130)( 86,131)( 87,132)( 88,133)( 89,134)( 90,135);
poly := sub<Sym(135)|s0,s1,s2>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

```
References : None.
to this polytope